Rapid Mixing Swendsen-Wang Sampler for Stochastic Partitioned Attractive Models

Abstract

The Gibbs sampler is a particularly popular Markov chain used for learning and inference problems in Graphical Models (GMs). These tasks are computationally intractable in general, and the Gibbs sampler often suffers from slow mixing. In this paper, we study the Swendsen-Wang dynamics which is a more sophisticated Markov chain designed to overcome bottlenecks that impede the Gibbs sampler. We prove O(log n) mixing time for attractive binary pairwise GMs (i.e., ferromagnetic Ising models) on stochastic partitioned graphs having n vertices, under some mild conditions, including low temperature regions where the Gibbs sampler provably mixes exponentially slow. Our experiments also confirm that the Swendsen-Wang sampler significantly outperforms the Gibbs sampler when they are used for learning parameters of attractive GMs.

Empirical Sampling of Connected Graph Partitions for Redistricting

“…This constraint connects the behavior of redistricting methods to known results about self-avoiding walks, but makes it more difficult to use intuitions from spin glass models. In particular, fast mixing properties of cluster-based spin glass models over portions of the parameter space [22,31,32,50] do not appear to hold generically for the samplers we consider. For instance, [48] demonstrates general obstructions to efficient uniform sampling from the space of connected districting plans, along with explicit families of graphs where the Glauber dynamics based Markov chains on districting plans mix slowly.…”